Arbitrary Order Fractional Difference Operators with Discrete

Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 4149320, 8 pages https://doi.org/10.1155/2017/4149320

Research Article Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications Thabet Abdeljawad,1 Qasem M. Al-Mdallal,2 and Mohamed A. Hajji2 1

Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al Ain, Abu Dhabi, UAE

2

Correspondence should be addressed to Thabet Abdeljawad; [email protected] Received 3 April 2017; Accepted 25 May 2017; Published 21 June 2017 Academic Editor: Garyfalos Papashinopoulos Copyright © 2017 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0 < 𝛼 ≤ 1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional difference operators to arbitrary positive order. The extension is given to both left and right fractional differences and sums. Then, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial difference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional difference boundary value problems of order 2 < 𝛼 ≤ 3 is proved and the ordinary difference Lyapunov inequality then follows as 𝛼 tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional difference calculus is given.

1. Introduction In the last few decades, the continuous and discrete fractional differential equations have received considerable interest due to their importance in many scientific fields; see, by way of example not exhaustive enumeration, [1–7]. In [8], the authors introduced a fractional derivative with an exponential kernel which tends to the ordinary derivative as 𝛼 tends to 1. More properties of this fractional derivative have been studied in [9], where the correspondent fractional integral operator was formulated. Then, the authors in [7] defined the left and right fractional derivatives with exponential kernel in the Riemann sense and formulated the right fractional derivatives in the sense of Caputo-Fabrizio with complete investigation to the correspondent fractional integrals and all the discrete versions with integration and summation by parts applied in the fractional and discrete fractional variational calculus. Then, very recently, the same authors proved an interesting monotonicity result in the sense of this new fractional difference calculus in [10]. In the same direction, for the purpose of providing more fractional derivatives with different nonsingular kernels, the authors in [11] defined a fractional operator with

Mittag-Leffler kernel and in [12, 13] the complete details and discrete versions have been studied. The exponential kernel fractional derivatives and hence their discrete counterparts are quite different from the Mittag-Leffler kernel fractional operators. For example, the integral operator corresponding to exponential kernel fractional derivatives consists of a multiple of the function 𝑓 added to a multiple of the integration of 𝑓, whereas the Mittag-Leffler kernel correspondent integral operator consists of a multiple of 𝑓 and a Riemann-Liouville fractional integral of the same order. Also, the monotonicity coefficient of the CFR fractional difference operator of order 0 < 𝛼 ≤ 1 is 𝛼 as shown in [10], whereas for the discrete Mittag-Leffler CFR operator is 𝛼2 as proven in [14]. Motivated, by what we mentioned above, we extend the order of fractional difference type operators with discrete exponential kernels to arbitrary positive order, prove existence and uniqueness theorems for the fractional initial value difference problems, and finally prove a Lyapunov type inequality for the CFR fractional difference operators of order 2 < 𝛼 ≤ 3. The ordinary discrete Lyapunov inequality is then confirmed as 𝛼 tends to 2 from the right not as in the case of the classical fractional difference as 𝛼 tends to 2 from the left [15]. For various fractional Lyapunov extensions we refer, for

2

Discrete Dynamics in Nature and Society

example, to [16–29]. All the authors there were motivated by the following theorem on ordinary Lyapunov inequality.

This is fractionalising of the 𝑛-iterated nabla sum

󸀠󸀠

𝑦 (𝑡) + 𝑞 (𝑡) 𝑦 (𝑡) = 0,

𝑡 ∈ (𝑎, 𝑏) , 𝑦 (𝑎) = 𝑦 (𝑏) = 0, (1)

has a nontrivial solution, where 𝑞 is a real continuous function; then 𝑏

4 󵄨 󵄨 ∫ 󵄨󵄨󵄨𝑞 (𝑠)󵄨󵄨󵄨 𝑑𝑠 > . 𝑏−𝑎 𝑎

Theorem 2 (see [26]). Let 𝑞(𝑡) be a nontrivial, continuous, and nonnegative function with period 𝜔 and let 𝜔

0

4 . 𝜔

𝑥 (𝑡) + 𝑞 (𝑡) 𝑥 (𝑡) = 0,

− ∞ < 𝑡 < ∞,

1 𝑏−1 𝛼−1 ∑ (𝑠 − 𝜌 (𝑡)) 𝑓 (𝑠) , Γ (𝛼) 𝑠=𝑡

(∇𝑏−𝛼 𝑓) (𝑡) =

Definition 4 (see [7, 10]). For 𝛼 ∈ (0, 1) and 𝑓 defined on N𝑎 , or 𝑏 N = {𝑏, 𝑏 − 1, . . .} in right case, we have the following definitions: (i) The left (nabla) CFC fractional difference is given by

(3) 𝛼

(CFC𝑎 ∇ 𝑓) (𝑡) =

𝐵 (𝛼) 𝑡 ∑ (∇ 𝑓) (𝑠) (1 − 𝛼)𝑡−𝜌(𝑠) 1 − 𝛼 𝑠=𝑎+1 𝑠 𝑡

= 𝐵 (𝛼) ∑ (∇𝑠 𝑓) (𝑠) (1 − 𝛼)

(4)

For the classical fractional calculus which is behind many extensions, we refer the reader to [32–35] and for the sake of comparison with the classical discrete fractional case we refer to [36] and the references therein. In addition, for the discrete fractional operators and their duality we refer to [37–39]. The article will be organised as follows: In the remaining part of this section we shall give some basics about the discrete CFC and CFR fractional differences and their correspondent sums as used in [7, 10]. In Section 2, we extend the order of CFC and CFR fractional differences and their correspondent sums to arbitrary positive order and give some illustrative examples. In Section 3, we prove some existence and uniqueness theorems by means of Banach fixed point theorem and give some illustrative examples. In Section 4, we prove a Lyapunov type inequality for a fractional CFR difference boundary value problem of order 2 < 𝛼 ≤ 3 and give an application to the fractional difference Sturm-Liouville Eigenvalue problem (SLEP) to enrich the applicability of our proven Lyapunov inequality in the frame of fractional difference operators with discrete exponential kernels.

(8)

.

(ii) The right (nabla) CFC fractional difference has the following form: 𝛼

(CFC ∇𝑏 𝑓) (𝑡) =

(5)

(9)

= 𝐵 (𝛼) ∑ (−Δ 𝑠 𝑓) (𝑠) (1 − 𝛼)𝑠−𝑡 . 𝑠=𝑡

(iii) The left (nabla) CFR fractional difference is written as 𝛼

(CFR𝑎 ∇ 𝑓) (𝑡) =

𝑡 𝐵 (𝛼) ∇𝑡 ∑ 𝑓 (𝑠) (1 − 𝛼)𝑡−𝜌(𝑠) 1 − 𝛼 𝑠=𝑎+1 𝑡

= 𝐵 (𝛼) ∇𝑡 ∑ 𝑓 (𝑠) (1 − 𝛼)

𝑡−𝑠

(10)

.

𝑠=𝑎+1

(iv) The right (nabla) CFR fractional difference is given by 𝛼

Definition 3 (see [36]). For 𝛼 > 0, 𝑎 ∈ R, 𝜌(𝑠) = 𝑠 − 1, and 𝑓 a real-valued function defined on N𝑎 = {𝑎, 𝑎 + 1, . . .}, the left Riemann-Liouville fractional sum of order 𝛼 > 0 is defined by

𝐵 (𝛼) 𝑏−1 ∑ (−Δ 𝑠 𝑓) (𝑠) (1 − 𝛼)𝑠−𝜌(𝑡) 1 − 𝛼 𝑠=𝑡 𝑏−1

(CFR ∇𝑏 𝑓) (𝑡) =

2. Preliminaries

𝑡 1 𝛼−1 ∑ (𝑡 − 𝜌 (𝑠)) 𝑓 (𝑠) . Γ (𝛼) 𝑠=𝑎+1

𝑡−𝑠

𝑠=𝑎+1

are purely imaginary with modulus one.

(𝑎 ∇−𝛼 𝑓) (𝑡) =

(7)

where 𝑡𝛼 = Γ(𝑡 + 𝛼)/Γ(𝑡) and Γ(𝑡) is the well-known gamma special function of 𝑡.

Then the roots of the characteristic equation corresponding to Hills equation 󸀠󸀠

(6)

The right fractional integral ending at 𝑏, where usually we assume that 𝑎 ≡ 𝑏 (mod 1), is defined by

(2)

Notice that inequality (2) is known as the classical Lyapunov inequality. It is worth mentioning that Cheng [31] had pointed out that Lyapunov neither stated nor proved Theorem 1 but he only stated the following result.

∫ 𝑞 (𝑠) 𝑑𝑠 ≤

𝑡 1 𝑛−1 ∑ (𝑡 − 𝜌 (𝑠)) 𝑓 (𝑠) . (𝑛 − 1)! 𝑎+1

(𝑎 ∇−𝑛 𝑓) (𝑡) =

Theorem 1 (see [30]). If the boundary value problem

𝑏−1 𝐵 (𝛼) (−Δ 𝑡 ) ∑ 𝑓 (𝑠) (1 − 𝛼)𝑠−𝜌(𝑡) 1−𝛼 𝑠=𝑡 𝑏−1

(11)

= 𝐵 (𝛼) (−Δ 𝑡 ) ∑ 𝑓 (𝑠) (1 − 𝛼)𝑠−𝑡 , 𝑠=𝑡

where 𝐵(𝛼) is a normalization positive constant depending on 𝛼 satisfying 𝐵(0) = 𝐵(1) = 1, (∇𝑔)(𝑡) = 𝑔(𝑡) − 𝑔(𝑡 − 1), and (Δ𝑔)(𝑡) = 𝑔(𝑡 + 1) − 𝑔(𝑡).

Discrete Dynamics in Nature and Society −𝛼

3 𝛼

In [7, 8], it was verified that (CF𝑎 ∇ CFR𝑎 ∇ 𝑓)(𝑡) = 𝑓(𝑡) 𝛼 −𝛼 and (CFR𝑎 ∇ CF𝑎 ∇ 𝑓)(𝑡) = 𝑓(𝑡). Also, in the right case 𝛼 𝛼 𝛼 −𝛼 (CF 𝐼𝑏 CFR ∇𝑏 𝑓)(𝑡) = 𝑓(𝑡) and (CFR ∇𝑏 CF ∇𝑏 𝑓)(𝑡) = 𝑓(𝑡). From [7, 8] we recall the relation between the CFC and CFR fractional differences as 𝛼

𝛼

(CFC𝑎 ∇ 𝑓) (𝑡) = (CFR𝑎 ∇ 𝑓) (𝑡) −

𝐵 (𝛼) 𝑓 (𝑎) (1 − 𝛼)𝑡−𝑎 , 1−𝛼

(12)

−1

noting that (CF𝑎 ∇ 𝑓)(𝑡) = (𝑎 ∇−1 𝑓)(𝑡), we see that for 𝛼 = 𝑛+1 −𝛼 we have (CF𝑎 ∇ 𝑓)(𝑡) = (𝑎 ∇−(𝑛+1) 𝑓)(𝑡). Also, for 0 < 𝛼 ≤ 1 we reobtain the concepts defined in Definition 4. Therefore, our generalization to higher order case is confirmed. Analogously, in the right case we have the following extension. Definition 8. Let 𝑛 < 𝛼 ≤ 𝑛 + 1 and 𝑓 be defined on N𝑎 ∩ 𝑏 N. Set 𝛽 = 𝛼 − 𝑛. Then 𝛽 ∈ (0, 1] and we define 𝛼

and for the right case by 𝛼

𝛼

(CFC ∇𝑏 𝑓) (𝑡) = (CFR ∇𝑏 𝑓) (𝑡) 𝐵 (𝛼) 𝑓 (𝑏) (1 − 𝛼)𝑏−𝑡 . − 1−𝛼

𝛼 (CFR ∇𝑏 𝑓) (𝑡)

(13)

Lemma 5 (see [7]). For 0 < 𝛼 < 1, we have −𝛼 𝐶𝐹𝐶 𝛼 𝑎 ∇ 𝑓) (𝑡)

−𝛼 𝛼 (𝐶𝐹 ∇𝑏 𝐶𝐹𝐶∇𝑏 𝑓) (𝑡)

−𝛼

= 𝑓 (𝑡) − 𝑓 (𝑎) , (14) = 𝑓 (𝑡) − 𝑓 (𝑏) .

Proposition 9. For 𝑓 defined on N𝑎 ∩ 𝑏 N and 𝑛 < 𝛼 ≤ 𝑛 + 1, we have 𝛼

𝛼

𝑛 times

(iii) (⊖ Δ 𝑓)(𝑡) = (−1) (Δ 𝑓)(𝑡).

𝛼

𝛼

(𝐶𝐹𝐶∇𝑏 𝑓) (𝑡) = (𝐶𝐹𝑅 ∇𝑏 𝑓) (𝑡)

3. Higher Order Fractional Differences and Sums

−

Definition 6. Let 𝑛 < 𝛼 ≤ 𝑛 + 1 and 𝑓 be defined on N𝑎 ∩ 𝑏 N. Set 𝛽 = 𝛼 − 𝑛 and define 𝛽

(CFC𝑎 ∇ 𝑓) (𝑡) = (CFC𝑎 ∇ ∇𝑛 𝑓) (𝑡) , 𝛽 (CFR𝑎 ∇ ∇𝑛 𝑓) (𝑡) .

(15)

−𝛽 (𝑎 ∇−𝑛 CF𝑎 ∇ 𝑓) (𝑡) .

𝐵 (𝛼) ( Δ𝑛 𝑓) (𝑏) (1 − 𝛼)𝑏−𝑡 . 1−𝛼 ⊖

Proposition 10. For 𝑢(𝑡) defined on N𝑎 ∩ 𝑏 N and for some 𝑛 ∈ N0 with 𝑛 < 𝛼 ≤ 𝑛 + 1, we have (i) (𝐶𝐹𝑅𝑎 ∇

= 𝑢(𝑡).

−𝛼 𝐶𝐹𝑅 𝛼 𝑎 ∇ 𝑢)(𝑡)

𝑘 𝑘 = 𝑢(𝑡) − ∑𝑛−1 𝑘=0 ((∇ 𝑢)(𝑎)/𝑘!)(𝑡 − 𝑎) .

−𝛼 𝐶𝐹𝐶 𝛼 𝑎 ∇ 𝑢)(𝑡)

= 𝑢(𝑡) − ∑𝑛𝑘=0 ((∇𝑘 𝑢)(𝑎)/𝑘!)(𝑡 − 𝑎)𝑘 .

(16)

(ii) (𝐶𝐹𝑎 ∇

Note that if we use the convention that (𝑎 ∇−0 𝑓)(𝑡) = 𝑓(𝑡), then for the case 0 < 𝛼 ≤ 1 we have 𝛽 = 𝛼 and hence we obtain Definition 4. Also, the convention (∇0 𝑓)(𝑡) = 𝑓(𝑡) 𝛼 𝛼 leads to (CFR𝑎 ∇ 𝑓)(𝑡) and (CFC𝑎 ∇ 𝑓)(𝑡) as in Definition 4 for 0 < 𝛼 ≤ 1.

(iii) (𝐶𝐹𝑎 ∇

Remark 7. In Definition 6, if we let 𝛼 = 𝑛 + 1 then 𝛽 = 1 and 𝛼 1 hence (CFR𝑎 ∇ 𝑓)(𝑡) = (CFR𝑎 ∇ ∇𝑛 𝑓)(𝑡) = (∇𝑛+1 𝑓)(𝑡). Also, by

(20)

Next proposition explains the action of the arbitrary −𝛼 order sum operator CF𝑎 ∇ on the arbitrary order CFR and CFC differences (and vice versa) and the action of the CFR difference on the CF correspondent sum operator.

𝛼 𝐶𝐹 −𝛼 𝑎 ∇ 𝑢)(𝑡)

The associated fractional sum is given by =

(19)

and for the right case

𝑛 times 𝑛 𝑛

−𝛼 (CF𝑎 ∇ 𝑓) (𝑡)

𝐵 (𝛼) 𝑛 (∇ 𝑓) (𝑎) (1 − 𝛼)𝑡−𝑎 , 1−𝛼

−

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⋅ ⋅ ⋅ Δ𝑓)(𝑡). (ii) (Δ 𝑓)(𝑡) = (ΔΔ

=

(18)

(𝐶𝐹𝐶𝑎∇ 𝑓) (𝑡) = (𝐶𝐹𝑅𝑎 ∇ 𝑓) (𝑡)

𝑛

𝛼 (CFR𝑎 ∇ 𝑓) (𝑡)

−𝛽

(CF ∇𝑏 𝑓) (𝑡) = (∇𝑏−𝑛 CF ∇𝑏 𝑓) (𝑡) .

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⋅ ⋅ ⋅ ∇𝑓)(𝑡). (i) (∇𝑛 𝑓)(𝑡) = (∇∇

𝛼

(17)

An immediate extension of (12) and (13) by using Definition 6 is the following.

Notation. For a positive integer 𝑛, we have

𝑛

=

CFR 𝛽 ∇𝑏 (⊖ Δ𝑛 𝑓) (𝑡) .

The associated fractional integral is given by

Notice that we extend Definition 4 to arbitrary 𝛼 > 0 in the next section.

(𝐶𝐹𝑎 ∇

𝛽

(CFC ∇𝑏 𝑓) (𝑡) = CFC ∇𝑏 (⊖ Δ𝑛 𝑓) (𝑡) ,

Proof. (i) By Definition 6 and the statement after Definition 4, we have (CFR𝑎 ∇

𝛼 CF −𝛼 𝑎 ∇ 𝑢) (𝑡)

𝛽

−𝛽

= CFR𝑎 ∇ ∇𝑛 𝑎 ∇−𝑛 CF𝑎 ∇ 𝑢 (𝑡) =

CFR 𝛽 CF −𝛽 𝑎 ∇ 𝑎 ∇ 𝑢 (𝑡)

= 𝑢 (𝑡) .

(21)

4

Discrete Dynamics in Nature and Society Notice that the condition 𝐹(𝑎) = 0 verifies the initial condition 𝑦(𝑎) = 𝑐. In addition, when 𝛼 → 1 we obtain the solution of the ordinary difference initial value problem (∇𝑦)(𝑡) = 𝐹(𝑡), 𝑦(𝑎) = 𝑐. (ii) Assume 1 < 𝛼 ≤ 2, 𝐹(𝑎) = 0, 𝑦(𝑎) = 𝑐1 , and −𝛼 (∇𝑦)(𝑎) = 𝑐2 . By applying CF𝑎 ∇ and making use of Proposition 10 and Definition 6 with 𝛽 = 𝛼 − 1, we obtain the solution

(ii) By Definition 6 and the statement after Definition 4, we have (CF𝑎 ∇

−𝛼 CFR 𝛼 𝑎 ∇ 𝑢) (𝑡)

−𝛽 CFR 𝛽 𝑛 𝑎 ∇ ∇ 𝑢) (𝑡)

= (𝑎 ∇−𝑛 CF𝑎 ∇

= 𝑎 ∇−𝑛 ∇𝑛 𝑢 (𝑡)

(22)

𝑛−1 (∇𝑘 𝑢) (𝑎)

= 𝑢 (𝑡) − ∑

𝑘!

𝑘=0

(𝑡 − 𝑎)𝑘 ,

𝑦 (𝑡) = 𝑐1 + 𝑐2 (𝑡 − 𝑎) +

where 𝛽 = 𝛼 − 𝑛. (iii) By Lemma 5 applied to 𝑓(𝑡) = (∇𝑛 𝑢)(𝑡), we have (CF𝑎 ∇

−𝛼 CFC 𝛼 𝑎 ∇ 𝑢) (𝑡)

𝑡 𝛼−1 + ∑ (𝑡 − 𝜌 (𝑠)) 𝐹 (𝑠) . 𝐵 (𝛼 − 1) 𝑠=𝑎+1

𝛽

= 𝑎 ∇−𝑛 𝑎 ∇−𝛽CFC𝑎 ∇ ∇𝑛 𝑢 (𝑡)

𝑛−1 (∇𝑘 𝑢) (𝑎)

𝑘!

𝑘=0

− (∇𝑛 𝑢) (𝑎) 𝑛

= 𝑢 (𝑡) − ∑

(23)

(∇𝑘 𝑢) (𝑎) 𝑘!

𝑘=0

(𝑡 − 𝑎)𝑘

(𝑡 − 𝑎)𝑛 𝑛!

In the next section, we prove existence and uniqueness theorems for some types of CFC and CFR initial value difference problems.

(𝑡 − 𝑎)𝑘 .

Example 13. Consider the CFC difference boundary value problem 𝛼

(CFC𝑎 ∇ 𝑦) (𝑡) + 𝑞 (𝑡) 𝑦 (𝑡) = 0,

Using the facts that ⊖ Δ𝑛 ∇𝑏−𝑛 𝑔(𝑡) = 𝑔(𝑡), 𝑛−1 ( Δ ⊖

∇𝑏𝑛 ⊖ Δ𝑛 𝑔 (𝑡) = 𝑔 (𝑡) − ∑

𝑘=0

𝑘

𝑔) (𝑏)

𝑘!

(𝑏 − 𝑡)𝑘 ,

1 < 𝛼 ≤ 2, 𝑡 ∈ N𝑎,𝑏 , 𝑦 (𝑎) = 𝑦 (𝑏) = 0. (24)

Proposition 11. For 𝑢(𝑡) defined on N𝑎,𝑏 and 𝛼 ∈ (𝑛, 𝑛 + 1], for some 𝑛 ∈ N0 , we have

Then 𝛽 = 𝛼 − 1 and by Proposition 10 if we apply the operator we obtain the solution −𝛼

𝑦 (𝑡) = 𝑐1 + 𝑐2 (𝑡 − 𝑎) − (CF𝑎 ∇ 𝑞 (⋅) 𝑦 (⋅)) (𝑡) . −𝛼

(CF𝑎 ∇ 𝑞 (⋅) 𝑦 (⋅)) (𝑡) =

−𝛼

−𝛼 𝐶𝐹𝑅 𝛼 ∇𝑏 𝑢)(𝑡) −𝛼 𝛼 (𝐶𝐹 ∇𝑏 𝐶𝐹𝐶∇𝑏 𝑢)(𝑡)

(iii)

Example 12. Consider the initial value problem: 𝛼

(CFC𝑎 ∇ 𝑦) (𝑡) = 𝐹 (𝑡) ,

(25)

(i) Assume 0 < 𝛼 ≤ 1, 𝑦(𝑎) = 𝑐, and 𝐹(𝑎) = 0. By −𝛼 applying CF𝑎 ∇ and making use of Proposition 10, we get the solution 𝑦 (𝑡) = 𝑐 +

𝑡

𝛼 1−𝛼 𝐹 (𝑡) + ∑ 𝐹 (𝑠) . 𝐵 (𝛼) 𝐵 (𝛼) 𝑠=𝑎+1

(26)

(30)

Hence, the solution has the form 𝑦 (𝑡) = 𝑐1 + 𝑐2 (𝑡 − 𝑎) −

where 𝐹(𝑡) is defined on N𝑎,𝑏 = N𝑎 ∩ 𝑏 N. Let us consider two cases depending on the order 𝛼 > 0:

1−𝛽 𝑡 ∑ 𝑞 (𝑠) 𝑦 (𝑠) 𝐵 (𝛽) 𝑠=𝑎+1 𝛽 ∇−2 𝑞 (𝑡) 𝑦 (𝑡) . + 𝐵 (𝛽) 𝑎

𝑘 𝑘 = 𝑢(𝑡)−∑𝑛−1 𝑘=0 ((⊖ Δ 𝑢)(𝑏)/𝑘!)(𝑏−𝑡) .

= 𝑢(𝑡)−∑𝑛𝑘=0 ((⊖ Δ𝑘 𝑢)(𝑏)/𝑘!)(𝑏−𝑡)𝑘 .

(29)

But

(i) (𝐶𝐹𝑅 ∇𝑏 𝐶𝐹 ∇𝑏 𝑢)(𝑡) = 𝑢(𝑡). (ii) (𝐶𝐹 ∇𝑏

(28)

CF −𝛼 𝑎∇ ,

and making use of Lemma 5 and the statement after Definition 4, we can state, for the right case, the following.

𝛼

(27)

Notice that the solution 𝑦(𝑡) verifies 𝑦(𝑎) = 𝑐1 without the use of 𝐹(𝑎) = 0. However, it verifies (∇𝑦)(𝑎) = 𝑐2 under the assumption 𝐹(𝑎) = 0. Also, note that when 𝛼 → 2 we recover the solution of the second-order ordinary difference initial value problem (∇2 𝑦)(𝑡) = 𝐹(𝑡), 𝑦(𝑎) = 𝑐1 , and (∇𝑦)(𝑎) = 𝑐2 .

= 𝑎 ∇−𝑛 [∇𝑛 𝑢 (𝑡) − (∇𝑛 𝑢) (𝑎)] = 𝑢 (𝑡) − ∑

𝑡 2−𝛼 ∑ 𝐾 (𝑠) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

𝑡 2−𝛼 ∑ 𝑞 (𝑠) 𝑦 (𝑠) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

𝑡 𝛼−1 − ∑ (𝑡 − 𝜌 (𝑠)) 𝑞 (𝑠) 𝑦 (𝑠) . 𝐵 (𝛼 − 1) 𝑠=𝑎+1

(31)

The boundary conditions imply that 𝑐1 = 0 and 𝑐2 =

𝑏 2−𝛼 ∑ 𝑞 (𝑠) 𝑦 (𝑠) (𝑏 − 𝑎) 𝐵 (𝛼 − 1) 𝑠=𝑎+1 𝑏 𝛼−1 + ∑ (𝑏 − 𝜌 (𝑠)) 𝑞 (𝑠) 𝑦 (𝑠) . (𝑏 − 𝑎) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

(32)


5 Proposition 10. The condition 𝑓(𝑎, 𝑦(𝑎)) = 0 always can not be avoided as we have seen in Example 12 with 𝑓(𝑡, 𝑦(𝑡)) = 𝐹(𝑡). As a result of Theorem 14, we conclude that the fractional difference linear initial value problem

Hence, 𝑦 (𝑡) =

(2 − 𝛼) (𝑡 − 𝑎) 𝑏 ∑ 𝑞 (𝑠) 𝑦 (𝑠) (𝑏 − 𝑎) 𝐵 (𝛼 − 1) 𝑠=𝑎+1 −

(𝛼 − 1) (𝑡 − 𝑎) ∑ (𝑏 − 𝜌 (𝑠)) 𝑞 (𝑠) 𝑦 (𝑠) (𝑏 − 𝑎) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

𝑡 2−𝛼 − ∑ 𝑞 (𝑠) 𝑦 (𝑠) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

−

𝛼

(CFC𝑎 ∇ 𝑦) (𝑡) = 𝑟𝑦 (𝑡) ,

𝑏

(33)

𝑡 𝛼−1 ∑ (𝑡 − 𝜌 (𝑠)) 𝑞 (𝑠) 𝑦 (𝑠) . 𝐵 (𝛼 − 1) 𝑠=𝑎+1

𝑟 ∈ R, 𝑡 ∈ N𝑎,𝑏 , 0 < 𝛼 ≤ 1, 𝑦 (𝑎) = 𝑐,

(38)

can have only the trivial solution unless 𝛼 = 1. Indeed, the solution satisfies 𝑦(𝑡) = 𝑐 + 𝑟((1 − 𝛼)/𝐵(𝛼))𝑦(𝑡) + (𝛼𝑟/𝐵(𝛼)) ∑𝑡𝑠=𝑎+1 𝑦(𝑠). This solution is only verified at 𝑎 if (1 − 𝛼)𝑦(𝑎) = 0. Theorem 16. Consider the system 𝛼

(𝐶𝐹𝑅𝑎 ∇ 𝑦) (𝑡) = 𝑓 (𝑡, 𝑦 (𝑡)) ,

4. Existence and Uniqueness Theorems for the Initial Value Problem Types

𝑡 ∈ N𝑎,𝑏 , 1 < 𝛼 ≤ 2, 𝑦 (𝑎) = 𝑐,

In this section we prove existence and uniqueness theorems for CFC and CFR type initial value problems. Theorem 14. Consider the system

𝑡 ∈ N𝑎,𝑏 , 0 < 𝛼 ≤ 1, 𝑦 (𝑎) = 𝑐,

(34)

𝐶𝐹 −𝛼 𝑎 ∇ 𝑓 (𝑡, 𝑦 (𝑡)) .

(35)

Proof. First, by the help of Proposition 10, (12), and taking into account the fact that 𝑓(𝑎, 𝑦(𝑎)) = 0, it is straight forward to prove that 𝑦(𝑡) satisfies system (34) if and only if it satisfies (35). Let 𝑋 = {𝑥 : max𝑡∈N𝑎,𝑏 |𝑥(𝑡)| < ∞} be the Banach space endowed with the norm ‖𝑥‖ = max𝑡∈N𝑎,𝑏 |𝑥(𝑡)|. On 𝑋 define the linear operator −𝛼

(𝑇𝑥) (𝑡) = 𝑐 + CF𝑎 ∇ 𝑓 (𝑡, 𝑥 (𝑡)) .

𝑦 (𝑡) = 𝑐 + 𝐶𝐹𝑎 ∇ 𝑓 (𝑡, 𝑦 (𝑡)) =𝑐+

such that 𝑓(𝑎, 𝑦(𝑎)) = 0, 𝐴((1 − 𝛼)/𝐵(𝛼) + 𝛼(𝑏 − 𝑎)/𝐵(𝛼)) < 1, and |𝑓(𝑡, 𝑦1 ) − 𝑓(𝑡, 𝑦2 )| ≤ 𝐴|𝑦1 − 𝑦2 |, 𝐴 > 0, where 𝑓 : N𝑎,𝑏 × R → R and 𝑦 : N𝑎,𝑏 → R. Then, system (34) has a unique solution of the form

(36)

Then, for arbitrary 𝑥1 , 𝑥2 ∈ 𝑋 and 𝑡 ∈ N𝑎,𝑏 , we have by assumption that 󵄨󵄨 󵄨 󵄨󵄨(𝑇𝑥1 ) (𝑡) − (𝑇𝑥2 ) (𝑡)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −𝛼 = 󵄨󵄨󵄨CF𝑎 ∇ [𝑓 (𝑡, 𝑥1 (𝑡)) − 𝑓 (𝑡, 𝑥2 (𝑡))]󵄨󵄨󵄨 (37) 󵄨 󵄨 ≤ 𝐴(

such that (𝐴/𝐵(𝛼−1))((2−𝛼)(𝑏−𝑎)+(𝛼−1)(𝑏−𝑎)2 /2) < 1, and |𝑓(𝑡, 𝑦1 )−𝑓(𝑡, 𝑦2 )| ≤ 𝐴|𝑦1 −𝑦2 |, 𝐴 > 0, where 𝑓 : N𝑎,𝑏 ×R → R and 𝑦 : N𝑎,𝑏 → R. Then, system (34) has a unique solution of the form −𝛼

𝛼

(𝐶𝐹𝐶𝑎∇ 𝑦) (𝑡) = 𝑓 (𝑡, 𝑦 (𝑡)) ,

𝑦 (𝑡) = 𝑐 +

(39)

1 − 𝛼 𝛼 (𝑏 − 𝑎) 󵄩󵄩 󵄩 + ) 󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 , 𝐵 (𝛼) 𝐵 (𝛼)

and hence 𝑇 is a contraction. By Banach fixed point theorem, there exists a unique 𝑥 ∈ 𝑋 such that 𝑇𝑥 = 𝑥 and hence the proof is complete. Remark 15. Similar existence and uniqueness theorems can be proved for system (34) with higher order by making use of

+

𝑡 2−𝛼 ∑ 𝑓 (𝑠, 𝑦 (𝑠)) 𝐵 (𝛼 − 1) 𝑠=𝑎+1

(40)

𝛼−1 ( ∇−2 𝑓 (⋅, 𝑦 (⋅)) (𝑡)) . 𝐵 (𝛼 − 1) 𝑎 −𝛼

Proof. If we apply CF𝑎 ∇ to system (39) and make use of Proposition 10 with 𝛽 = 𝛼 − 1 then we reach at the 𝛼 representation (40). Conversely, if we apply CFR𝑎 ∇ , make use of Proposition 10 and by noting that CFR 𝛼 𝑎∇

𝛽

= CFR𝑎 ∇ ∇𝑡 𝑐 = 0,

(41)

we obtain system (39). Hence, 𝑦(𝑡) satisfies system (39) if and only if it satisfies (40). Let 𝑋 = {𝑥 : max𝑡∈N𝑎,𝑏 |𝑥(𝑡)| < ∞} be the Banach space endowed with the norm ‖𝑥‖ = max𝑡∈N𝑎,𝑏 |𝑥(𝑡)|. On 𝑋 define the linear operator −𝛼

(𝑇𝑥) (𝑡) = 𝑐 + CF𝑎 ∇ 𝑓 (𝑡, 𝑥 (𝑡)) .

(42)

Then, for arbitrary 𝑥1 , 𝑥2 ∈ 𝑋 and 𝑡 ∈ N𝑎,𝑏 , we have by assumption that 󵄨󵄨 󵄨 󵄨󵄨(𝑇𝑥1 ) (𝑡) − (𝑇𝑥2 ) (𝑡)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 −𝛼 = 󵄨󵄨󵄨CF𝑎 ∇ [𝑓 (𝑡, 𝑥1 (𝑡)) − 𝑓 (𝑡, 𝑥2 (𝑡))]󵄨󵄨󵄨 󵄨 󵄨 (43) 𝐴 (𝛼 − 1) (𝑏 − 𝑎)2 ≤ ((2 − 𝛼) (𝑏 − 𝑎) + ) 𝐵 (𝛼 − 1) 2 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 , and hence 𝑇 is a contraction. By Banach Contraction Principle, there exists a unique 𝑥 ∈ 𝑋 such that 𝑇𝑥 = 𝑥 and hence the proof is complete.

6

Discrete Dynamics in Nature and Society Lemma 18. Given that 𝑏 ≡ 𝑎 (mod 1), the Green function 𝐺(𝑡, 𝑠) defined in Lemma 17 has the following properties:

5. The Lyapunov Inequality for the CFR Difference Boundary Value Problem In this section, we prove a Lyapunov inequality for a CFR boundary value difference problem of order 2 < 𝛼 ≤ 3. Consider the boundary value problem 𝛼 (CFR𝑎 ∇ 𝑦) (𝑡)

𝜌

+ 𝑞 (𝑡) 𝑦 (𝑡) = 0,

2 < 𝛼 ≤ 3, 𝑡 ∈ N𝑎+1,𝑏−1 , 𝑦 (𝑎) = 𝑦 (𝑏) = 0,

(44)

where, 𝑦𝜌 (𝑡) = 𝑦(𝜌(𝑡)) = 𝑦(𝑡 − 1). Lemma 17. 𝑦(𝑡) is a solution of the boundary value problem (44) if and only if it satisfies the equation 𝑏

𝑦 (𝑡) = ∑ 𝐺 (𝑡, 𝑠) 𝑇 (𝑠, 𝑦 (𝑠)) , where 𝐺 (𝑡, 𝑠)

𝑇 (𝑡, 𝑦 (𝑡)) = =

𝐶𝐹 𝛽 𝑎∇

𝑡 + 1 ≤ 𝑠, 𝑡, 𝑠 ∈ N𝑎,𝑏 , 𝑠 − 1 ≤ 𝑡, 𝑡, 𝑠 ∈ N𝑎,𝑏 ,

(46)

(𝑞 (⋅) 𝑦𝜌 (⋅)) (𝑡)

𝛽 1−𝛽 𝑞 (𝑡) 𝑦𝜌 (𝑡) + ( ∇−1 𝑞 (⋅) 𝑦𝜌 (⋅)) (𝑡) , 𝛽 = 𝛼 − 2. 𝐵 (𝛽) 𝐵 (𝛽) 𝑎 −𝛼

Proof. Apply the integral CF𝑎 ∇ to (44) and make use of Definition 6 and Proposition 10 with 𝑛 = 2 and 𝛽 = 𝛼 − 2 to reach 𝑦 (𝑡) = 𝑐1 + 𝑐2 (𝑡 − 𝑎) − (𝑎 ∇−2 𝑇 (⋅, 𝑦 (⋅))) (𝑡) 𝑡

= 𝑐1 + 𝑐2 (𝑡 − 𝑎) − ∑ (𝑡 − 𝜌 (𝑠)) 𝑇 (𝑠, 𝑦 (𝑠)) .

(47)

𝑠=𝑎+1

The condition 𝑦(𝑎) = 0 implies that 𝑐1 = 0 and the condition 𝑦(𝑏) = 0 implies that 𝑐2 = (1/(𝑏 − 𝑎)) ∑𝑏𝑠=𝑎+1 (𝑏 − 𝜌(𝑠))𝑇(𝑠, 𝑦(𝑠)), and hence 𝑦 (𝑡) =

𝑡−𝑎 𝑏 ∑ (𝑏 − 𝜌 (𝑠)) 𝑇 (𝑠, 𝑦 (𝑠)) 𝑏 − 𝑎 𝑠=𝑎+1 𝑡

(48)

− ∑ (𝑡 − 𝜌 (𝑠)) 𝑞 (𝑠) 𝑇 (𝑠, 𝑦 (𝑠)) 𝑑𝑠. 𝑠=𝑎+1

Then, the result follows by splitting the summation ∑ (𝑏 − 𝜌 (𝑠)) 𝑇 (𝑠, 𝑦 (𝑠)) 𝑡

𝑏

+ ∑ (𝑏 − 𝜌 (𝑠)) 𝑇 (𝑠, 𝑦 (𝑠)) . 𝑠=𝑡+1

𝑏−𝑎 (𝑎 + 𝑏 + 2) { )= { {𝑓 ( 2 4 ={ + 𝑏 + 3) − 𝑎)2 − 1 (𝑎 (𝑏 { {𝑓 ( )= 2 4 (𝑏 − 𝑎) {

if 𝑏 − 𝑎 is even (50) if 𝑏 − 𝑎 is odd.

Hence, in either cases max𝑠∈N𝑎,𝑏 𝑓(𝑠) ≤ (𝑏 − 𝑎)/4. Proof. (1) It is clear that 𝑔1 (𝑡, 𝑠) = (𝑡 − 𝑎)(𝑏 − 𝜌(𝑠))/(𝑏 − 𝑎) ≥ 0. Regarding the part 𝑔2 (𝑡, 𝑠) = ((𝑡−𝑎)(𝑏−𝜌(𝑠))/(𝑏−𝑎)−(𝑡−𝜌(𝑠))) we see that (𝑡 − 𝜌(𝑠)) = ((𝑡 − 𝑎)/(𝑏 − 𝑎))(𝑏 − (𝑎 + (𝜌(𝑠) − 𝑎)(𝑏 − 𝑎)/(𝑡 − 𝑎))) and that 𝑎 + (𝜌(𝑠) − 𝑎)(𝑏 − 𝑎)/(𝑡 − 𝑎) ≥ 𝜌(𝑠) if and only if 𝜌(𝑠)(𝑏 − 𝑡) + 𝑎(𝑡 − 𝑏) ≥ 0 if and only if 𝑠 ≥ 𝑎 + 1. Hence, we conclude that 𝑔2 (𝑡, 𝑠) ≥ 0 as well. (2) Clearly, 𝑔1 (𝑡, 𝑠) is an increasing function in 𝑡. Applying ∇ to 𝑔2 with respect to 𝑡 for every fixed 𝑠 ≥ 𝑎 + 1 we see that ∇𝑡 𝑔(𝑡, 𝑠) = (𝑏 − 𝑎)−1 (𝑎 − 𝜌(𝑠)) and hence 𝑔2 is a decreasing function in 𝑡. (3) Let 𝑓(𝑠) = 𝐺(𝜌(𝑠), 𝑠) = (𝜌(𝑠) − 𝑎)(𝑏 − 𝜌(𝑠))/(𝑏 − 𝑎). Then, 𝑎 + 𝑏 − 2𝑠 + 3 (51) = 0, 𝑏−𝑎 if 𝑠 = (𝑎 + 𝑏 + 3)/2 and hence for 𝑎, 𝑏 ∈ N, 𝑓 attains its maximum at 𝑠1 = (𝑎 + 𝑏 + 3)/2 if 𝑎 + 𝑏 (or 𝑏 − 𝑎) is odd and at (𝑎 + 𝑏 + 2)/2 if 𝑎 + 𝑏 (or 𝑏 − 𝑎) is even. More generally, if 𝑎 and 𝑏 are such that 𝑏 ≡ 𝑎 (mod 1), we see that 𝑠1 ≡ 𝑎 (mod 1) if 𝑏 − 𝑎 is odd and 𝑠2 ≡ 𝑎 (mod 1) if 𝑏 − 𝑎 is even. Finally, 𝑓(𝑠1 ) = ((𝑏 − 𝑎)2 − 1)/4(𝑏 − 𝑎) and 𝑓(𝑠2 ) = (𝑏 − 𝑎)/4. (∇𝑓) (𝑠) =

In next lemma, we estimate 𝑇(𝑡, 𝑦(𝑡)) for a function 𝑦 ∈ 𝐵[N𝑎,𝑏 ], the Banach space of all Banach-valued finite sequences on N𝑎,𝑏 with ‖𝑦‖ = max𝑡∈N𝑎,𝑏 |𝑦(𝑡)|, where | ⋅ | is the norm in the Banach space. Lemma 19. For 𝑦 ∈ 𝐵[N𝑎,𝑏 ] and 2 < 𝛼 ≤ 3, 𝛽 = 𝛼 − 2, we have, for any 𝑡 ∈ N𝑎,𝑏 , 󵄨󵄨 󵄨 󵄩 󵄩 (52) 󵄨󵄨𝑇 (𝑡, 𝑦 (𝑡))󵄨󵄨󵄨 ≤ 𝑅 (𝑡) 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 ,

𝑅 (𝑡) = [

𝑠=𝑎+1

𝑠=𝑎+1

max 𝑓 (𝑠)

𝑠∈N𝑎,𝑏

where

𝑏

= ∑ (𝑏 − 𝜌 (𝑠)) 𝑇 (𝑠, 𝑦 (𝑠))

(3) 𝑓(𝑠) = 𝐺(𝜌(𝑠), 𝑠) = (𝜌(𝑠) − 𝑎)(𝑏 − 𝜌(𝑠))/(𝑏 − 𝑎) has a unique maximum, given by

(45)

𝑠=𝑎+1

(𝑡 − 𝑎) (𝑏 − 𝜌 (𝑠)) { , { { 𝑏−𝑎 ={ { {( (𝑡 − 𝑎) (𝑏 − 𝜌 (𝑠)) − (𝑡 − 𝜌 (𝑠))) , 𝑏−𝑎 {

(1) 𝐺(𝑡, 𝑠) ≥ 0 for all 𝑡, 𝑠 ∈ N𝑎,𝑏 . (2) max𝑡∈N𝑎,𝑏 𝐺(𝑡, 𝑠) = 𝐺(𝜌(𝑠), 𝑠) for 𝑠 ∈ N𝑎+1,𝑏−1 .

(49)

𝑡 3 − 𝛼 󵄨󵄨 𝛼−2 󵄨 󵄨 󵄨 ∑ 󵄨󵄨󵄨𝑞 (𝑠)󵄨󵄨󵄨] . 󵄨󵄨𝑞 (𝑡)󵄨󵄨󵄨 + 𝐵 (𝛼 − 2) 𝐵 (𝛼 − 2) 𝑠=𝑎+1

(53)

Theorem 20 (the CFR fractional difference Lyapunov inequality). If the boundary value problem (44) has a nontrivial solution, where 𝑞(𝑡) is a real-valued bounded function on N𝑎,𝑏 , then 𝑏

∑ 𝑅 (𝑠) > 𝑠=𝑎+1

4 . 𝑏−𝑎

(54)


7

Proof. Assume 𝑦 ∈ 𝑌 = 𝐵[N𝑎,𝑏 ] is a nontrivial solution of the boundary value problem (44). By Lemma 17, 𝑦 must satisfy 𝑏

𝑦 (𝑡) = ∑ 𝐺 (𝑡, 𝑠) 𝑇 (𝑠, 𝑦 (𝑠)) .

(55)

𝑠=𝑎+1

Then, by using the properties of the Green function 𝐺(𝑡, 𝑠) proved in Lemmas 18 and 19, we come to the conclusion that 𝑏 󵄩 󵄩 󵄩󵄩 󵄩󵄩 𝑏 − 𝑎 ∑ 𝑅 (𝑠) 󵄩󵄩󵄩𝑦󵄩󵄩󵄩 󵄩󵄩𝑦󵄩󵄩 < 4 𝑠=𝑎+1

(56)

from which (54) follows. Remark 21. Note that if 𝛼 → 2+ , then 𝑅(𝑡) tends to |𝑞(𝑡)| and hence we obtain the classical nabla discrete version of the Lyapunov inequality (2). For the sake of more comparisons of Lyapunov inequalities on time scales we refer to [40]. Example 22. Consider the following CFR Sturm-Liouville difference eigenvalue problem (SLDEP) of order 2 < 𝛼 ≤ 3

𝑡 ∈ N1,𝑏−1 , 𝑏 ∈ N, 𝑦 (0) = 𝑦 (𝑏) = 0.

(57)

If 𝜆 is an eigenvalue of (57), then by Theorem 20 with 𝑞(𝑡) = 𝜆, we have 𝑇 (𝑡) = [

3−𝛼 𝛼−2 ( ∇−1 |𝜆|) (𝑡)] |𝜆| + 𝐵 (𝛼 − 2) 𝐵 (𝛼 − 2) 0

= |𝜆| [

3−𝛼 𝛼−2 + 𝑡] . 𝐵 (𝛼 − 2) 𝐵 (𝛼 − 2)

𝑏

𝑠=1

𝑏 (3 − 𝛼) 𝑏2 (𝛼 − 2) 4 + ]> . 𝐵 (𝛼 − 2) 2𝐵 (𝛼 − 2) 𝑏

(58)

(59)

Notice that the limiting case 𝛼 → 2+ implies that |𝜆| > 4/𝑏2 which is the lower bound for the eigenvalues of the ordinary difference eigenvalue problem: (∇2 𝑦) (𝑡) + 𝜆𝑦𝜌 (𝑡) = 0, 𝑡 ∈ N1,𝑏−1 , 𝑦 (0) = 𝑦 (𝑏) = 0.

The authors declare that they have no conflicts of interest.

The first author would like to thank the Research and Translation Center in Prince Sultan University for continuous encouragement and support.

References

Hence, we must have ∑𝑇 (𝑠) = |𝜆| [

Conflicts of Interest

Acknowledgments

𝛼

(CFR0 ∇ 𝑦) (𝑡) + 𝜆𝑦𝜌 (𝑡) = 0,

achieved. To set up the basic concepts, we proved existence and uniqueness theorems by means of Banach fixed point theorem for initial value problems in the frame of CFC and CFR fractional differences. We have come to the conclusion that the condition 𝑓(𝑎, 𝑦(𝑎)) = 0 is necessary to guarantee the existence of solution and hence fractional linear difference initial value problem with constant coefficients results in the trivial solution unless the order is positive integer. We used our extension to arbitrary order to prove a Lyapunov type inequality for a CFR boundary value problem of order 2 < 𝛼 ≤ 3 and then obtain the classical ordinary case when 𝛼 tends to 2 from right. This proves different behavior from the classical fractional difference case, where the Lyapunov inequality was proved for a fractional difference boundary problem of order 1 < 𝛼 ≤ 2 and the classical ordinary case was then recovered when 𝛼 tends to 2 from left.

(60)

6. Conclusions Fractional differences and their correspondent fractional sum operators are of importance in discrete modeling of various problems in science. We extended the fractional difference calculus whose difference operators depend on a discrete exponential function kernel to arbitrary positive order. The correspondent arbitrary order fractional sum operators have been defined as well and applied to solve fractional initial and boundary value difference problems. The extension for right fractional differences and sums is also

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